Every complete graph on at least three vertices is evidently Hamiltonian, as the vertices of a Hamilton cycle can be selected one by one in an arbitrary order.
Conditions to guarantee the existence of a Hamilton cycle have
been studied actively.
In particular, we may
ask how large the minimum degree can be in order to guarantee the existence of
a Hamilton cycle. The celebrated Dirac theorem
[1]} answers this question.
It states that every graph with \(n\ge 3\)  vertices and minimum degree at least \(n/2\)  has
a Hamilton cycle.
The condition is sharp when we consider the complete bipartite
graph with the parts of sizes \(\lfloor \frac{n-1}{2}\rfloor \)  and
\(\lfloor \frac{n+1}{2}\rfloor \) .
